A Simpler Solution

I’m guessing this was about 5 years ago. I was at an all-day workshop when a high school math teacher, sitting next to me, asked about the PoW (from mathforum.org) that I had assigned to my students. I happened to have an extra copy in my backpack and gave it to her.

Dad’s Cookies [Problem #2959]

Dad bakes some cookies. He eats one hot out of the oven and leaves the rest on the counter to cool. He goes outside to read.

Dave comes into the kitchen and finds the cookies. Since he is hungry, he eats half a dozen of them.

Then Kate wanders by, feeling rather hungry as well. She eats half as many as Dave did.

Jim and Eileen walk through next, each of them eats one third of the remaining cookies.

Hollis comes into the kitchen and eats half of the cookies that are left on the counter.

Last of all, Mom eats just one cookie.

Dad comes back inside, ready to pig out. “Hey!” he exclaims, “There is only one cookie left!”

How many cookies did Dad bake in all?

Maybe you’d like to work on this problem before reading on.

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The teacher started solving the problem. She was really into it, so much so that I felt she’d ignored much of what our presenter was presenting at the time. She ran out of paper and grabbed some more. She looked up from her papers at one point and said something that I interpreted as I-know-this-problem-is-not-that-hard-but-what-the-fuck.

It was now morning break.

She worked on it some more.

By lunch time, she asked, “Okay, how do you solve this?” I read the problem again and drew some boxes on top of the paper that she’d written on. (Inside the green.)

She knew I’d solved the problem with a few simple sketches because she understood the drawings and what they represented. I just really appreciated her perseverance.

I share this with you because a few nights ago I was at our local Math Teachers’ Circle where Joshua Zucker led us through some fantastic activities with Zome models. We were asked for the volume of various polyhedrons relative to one another. Our group really struggled on one of the shapes. We used formulas and equations only to get completely befuddled, and our work ended up looking like one of the papers above.

Over the years I’ve heard a few students tell me, “Mrs. Nguyen, my uncle is an engineer, and he can’t help me with the PoW.” Substitute uncle with another grown-up family member. Substitute engineer with another profession, including math teacher. I remember getting a note from one of my student’s tutor letting me know that I shouldn’t be giving 6th graders problems that he himself cannot solve. (The student’s parent fired him upon learning this.)

I like to think that my love of problem solving will rub off on my kids. I hope they will love the power of drawing rectangles as much I do. Or just a tiny little bit.

8 Comments

I couldn’t read your stuff in the green loop, but I worked backwards from the 1 cookie, getting 16, and on a closer reading (!) I got 22. No paper or pencil needed at all. I like this, it shows that problem solvers try desperately to fit the problem to their preferred (or only) solution method. I once met a management scientist (operational researcher) who was an expert in linear programming. Every management or organisational problem he was hired to attack he poked and prodded an pushed it until it was formulated as a linear programming problem, which of course could be solved. The results obtained were not always solutions to the original problem !

Dad bakes some cookies. He eats one hot out of the oven and leaves the rest on the counter to cool. He goes outside to read.
He ate 1
Dave comes into the kitchen and finds the cookies. Since he is hungry, he eats half a dozen of them.
he ate 6, so 7 gone
Then Kate wanders by, feeling rather hungry as well. She eats half as many as Dave did.
She ate 3, so 7+3=10 gone
But the next step says that there were still 12 there ****************
Jim and Eileen walk through next, each of them eats one third of the remaining cookies.
They ate 2/3 of the cookies, leaving 4, so there were 12
Hollis comes into the kitchen and eats half of the cookies that are left on the counter.
He ate half, leaving 2, so there were 4
Last of all, Mom eats just one cookie.
So there were 2, and she ate 1, leaving 1

Dad comes back inside, ready to pig out. “Hey!” he exclaims, “There is only one cookie left!”

How many cookies did Dad bake in all?
************* after kate 10 had gone and there were still 12 : total 22

Yes, my work for this problem looked exactly like this (written down on my yellow notepad):

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1
2
4
12 (dubious phrasing)
15
21
22

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“Dubious phrasing” refers to the fact that I thought it was a little unclear whether Jim and Eileen both ate the same number of cookies, or if one of them ate 1/3 of what was there, and then the other ate 1/3 of what was left. But the phrasing seemed to suggest the former a little more clearly.

There are a number of problems like this in some of Raymond Smullyan’s riddle/puzzle books, and they’re a great example of why “work backward” is an important strategy to have in your toolkit (one of the ones suggested by Polya, incidentally).

I agree about the dubious phrasing for Jim and Eileen, but at least it’s solvable that way too. If Eileen ate 1/3 of what was left after Jim came through and ended up with 4, Jim must have left her 6, and if he ate 1/3 and had 6 left, there must have been 9 when he walked in. So compared to the “they both walked in together and ate 2/3 of what was there” option, all the other steps working backwards are 3 fewer, meaning 19 cookies to start.

There’s some nice stuff on similar problems at NCTM Illuminations, under http://illuminations.nctm.org/Lesson.aspx?id=1037 (in case that doesn’t work, they’re the Mangoes Problem and Sailors and Coconuts), with Guess & Check, Draw a Picture, and Work Backwards strategies explained along with the algebraic solution.

Wording is unambiguous. Jim ate a third of the remaining cookies. Eileen also ate one third of the cookies that remained when she walked through.
Starting with Dad – at the end – we have 1, 2, 4, 6, 9, 12 18, 19. So that first hot cookie Dad ate before he went off to read his paper was one of 19.

There are a couple of extensions for this that can bring the problem into algebra in a more open middle way. One is to give a different end point to different students/groups: what if Dad got 2 cookies? What if he got 0 or 3 cookies? Collect all that together and what do you notice?
Or you could use x for the last number Dad ends up with, and work backwards with algebraic notation (or – back to rectangles – bar models). In both cases you can come up with a function that poses its own curious riddle: what the heck do those numbers have to do with the problem?

I try to help the students accept that rectangles are tools, algebra is a tool, bar models are tools, and no tool is “the best”… they all have their strengths, problems in which they shine. I keep drawing little pictures whenever I share my thinking and eventually, sometimes a year and half later, many of them start drawing little pictures too.